left null space

$\operatorname{Nul}(A^{\mathsf T})$ — vectors $\mathbf y$ with $\mathbf y^{\mathsf T}A=\mathbf 0$ .

For an $m\times n$ matrix $A$ , the left null space is the set of $\mathbf y\in\mathbb R^m$ with $\mathbf y^{\mathsf T}A=\mathbf 0^{\mathsf T}$ . Transposing shows this is exactly $\operatorname{Nul}(A^{\mathsf T})$ , the ordinary null space of the transpose — so it requires no new theory, just row-reducing $A^{\mathsf T}$ .